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        <title>Delocalization of eigenvectors of random matrices with independent entries</title>
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<meta name="citation_abstract" content="We prove that an n&#215;n random matrix G with independent entries is completely delocalized. Suppose that the entries of G have zero means, variances uniformly bounded below, and a uniform tail decay of exponential type. Then with high probability all unit eigenvectors of G have all coordinates of magnitude O(n−1/2), modulo logarithmic corrections. This comes as a consequence of a new, geometric approach to delocalization for random matrices."  />
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<meta content="We prove that an n&#215;n random matrix G with independent entries is completely delocalized. Suppose that the entries of G have zero means, variances uniformly bounded below, and a uniform tail decay of exponential type. Then with high probability all unit eigenvectors of G have all coordinates of magnitude O(n−1/2), modulo logarithmic corrections. This comes as a consequence of a new, geometric approach to delocalization for random matrices."  property="og:description" />
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<meta name="dc.Creator" content="Mark Rudelson"  />
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<meta name="dc.Description" content="We prove that an n&#215;n random matrix G with independent entries is completely delocalized. Suppose that the entries of G have zero means, variances uniformly bounded below, and a uniform tail decay of exponential type. Then with high probability all unit eigenvectors of G have all coordinates of magnitude O(n−1/2), modulo logarithmic corrections. This comes as a consequence of a new, geometric approach to delocalization for random matrices."  />
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            <text class="DetailDate"><span>1 October 2015</span></text>
            <text class="ProceedingsArticleOpenAccessHeaderText">Delocalization of eigenvectors of random matrices with independent entries<br /></text>
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                            <span class="linkBlueToBlue" tabindex="0" onkeypress="if (event.which == 13) { this.click() }" onclick='window.location.href=&quot;/search?author=Mark_Rudelson&quot;'>Mark Rudelson</span>,
                            <span class="linkBlueToBlue" tabindex="0" onkeypress="if (event.which == 13) { this.click() }" onclick='window.location.href=&quot;/search?author=Roman_Vershynin&quot;'>Roman Vershynin</span>
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                        <span>Duke Math. J. </span>
                                            <span>164(13): </span>
                                            <span>2507-2538 </span>
                                            <span>(1 October 2015). </span>
                                            <span>DOI: 10.1215/00127094-3129809</span>
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<!-- *** Custom HTML *** --><p> We prove that an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="$n\times n$" overflow="scroll"><mi>n</mi><mo>×</mo><mi>n</mi></math> random matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="$G$" overflow="scroll"><mi>G</mi></math> with independent entries is completely delocalized. Suppose that the entries of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="$G$" overflow="scroll"><mi>G</mi></math> have zero means, variances uniformly bounded below, and a uniform tail decay of exponential type. Then with high probability all unit eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="$G$" overflow="scroll"><mi>G</mi></math> have all coordinates of magnitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="$O(n^{-1/2})$" overflow="scroll"><mi>O</mi><mo>(</mo><msup><mi>n</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></math>, modulo logarithmic corrections. This comes as a consequence of a new, geometric approach to delocalization for random matrices. </p>
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                                <span class="linkBlueToBlue" tabindex="0" onkeypress="if (event.which == 13) { this.click() }" onclick='window.location.href="/search?author=Mark_Rudelson"'>Mark Rudelson.</span>
                                <span class="linkBlueToBlue" tabindex="0" onkeypress="if (event.which == 13) { this.click() }" onclick='window.location.href="/search?author=Roman_Vershynin"'>Roman Vershynin.</span>
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                    <span>"Delocalization of eigenvectors of random matrices with independent entries."</span>
                    <span>Duke Math. J.</span>
                        <span>164</span>
                    <span>(13)</span>
                        <span>2507 - 2538,</span>
                                            <span>1 October 2015.</span>
                                            <span>https://doi.org/10.1215/00127094-3129809</span>
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                            <span>Received: 21 August 2013</span>;                                                     <span>Revised: 10 October 2014</span>;                                                                             <span>Published: 1 October 2015</span>
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                            First available in Project Euclid: 5 October 2015
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                                    <div class="reference-id">zbMATH: <a href="https://zbmath.org/?q=an:1352.60007">1352.60007</a></div>
                                    <div class="reference-id">MathSciNet: <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=MR3405592">MR3405592</a></div>
                                                    <div>Digital Object Identifier: 10.1215/00127094-3129809</div>
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                                        delocalization of eigenvectors
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        Mark Rudelson, Roman Vershynin "Delocalization of eigenvectors of random matrices with independent entries," Duke Mathematical Journal, Duke Math. J. 164(13), 2507-2538, (1 October 2015)
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